MinT - Best Known (19, 19+35, s)-Nets in Base 27

Best Known (19, 19+35, s)-Nets in Base 27

(19, 19+35, 108)-Net over F₂₇ — Constructive and digital

Digital (19, 54, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 54, 108)-net over F₂₇, using
- net from sequence [i] based on digital (18, 107)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 108, using
    - F₃ from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F₂₇ [i]

(19, 19+35, 148)-Net over F₂₇ — Digital

Digital (19, 54, 148)-net over F₂₇, using

t-expansion [i] based on digital (18, 54, 148)-net over F₂₇, using
- net from sequence [i] based on digital (18, 147)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 148, using
    - a curve from the manYPoints database [i]

(19, 19+35, 160)-Net in Base 27 — Constructive

(19, 54, 160)-net in base 27, using

2 times m-reduction [i] based on (19, 56, 160)-net in base 27, using
- base change [i] based on digital (5, 42, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(19, 19+35, 167)-Net in Base 27

(19, 54, 167)-net in base 27, using

2 times m-reduction [i] based on (19, 56, 167)-net in base 27, using
- base change [i] based on digital (5, 42, 167)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 166)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 167, using
      - a curve from the manYPoints database [i]

(19, 19+35, 7998)-Net in Base 27 — Upper bound on s

There is no (19, 54, 7999)-net in base 27, because

1 times m-reduction [i] would yield (19, 53, 7999)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 7295 689542 147483 833376 794828 600738 400385 259234 152675 489662 557459 156190 825159 > 27⁵³ [i]